SOME RESULTS ASSOCIATED WITH FRACTIONAL INTEGRALS INVOLVING THE EXTENDED CHEBYSHEV FUNCTIONAL
Zoubir DAHMANI
Abstract. In this paper, the Riemann-Liouville fractional integral is used to establish a new class of inequalities for the extended Chebyshev functional.
2000 Mathematics Subject Classification: 26D10, 26A33. 1. Introduction
Let us consider the functional [3] T (f, g, p, q) := Z b a q(x)dx Z b a p(x)f (x)g (x) dx + Z b a p(x)dx Z b a q(x)f (x) g (x) dx −Rb aq(x)f (x) dx Rb ap(x)g (x) dx −Rb ap(x)f (x) dx Rb aq(x)g (x) dx , (1) where f and g are two integrable functions on [a, b] and p, q are two positive inte-grable functions on [a, b].
In the case of f0, g0 ∈ L∞(a, b), S.S. Dragmir [9] proved that
|S(f, g, p)| ≤ ||f0||∞||g0||∞ hZ b a p(x)dx Z b a x2p(x)dx − Z b a xp(x)dx2i, (2) where S(f, g, p) := 1 2T (f, g, p, p) = Z b a p (x) Z b a p(x)f (x) g (x) dx −Rb ap(x)f (x) dx Rb ap(x)g (x) dx. (3) If f is M -g-Lipschitzian on [a, b] : i.e.
Dragomir proved that |S(f, g, p)| ≤ Mh Z b a p(x)dx Z b a g2(x)p(x)dx − Z b a g(x)p(x)dx2i. (5) Many researchers have given considerable attention to (1) and (3) and several in-equalities related to these functionals have appeared in the literature, to mention a few, see [1,2,4-12,14-18]and the references cited therein.
The main aim of this paper is to establish some new generalizations for the extended Chebyshev functional (1) by using the Riemann-Liouville fractional integrals. We give our results in the case of differentiable functions whose derivative belong to L∞([0, ∞[. Then, under the condition (4), we give another class of inequalities.
2. Basic Definitions of the Fractional Integrals
Definition 1. A real valued function f (t), t ≥ 0 is said to be in the space Cµ, µ ∈ IR, if there exists a real number p > µ such that f (t) = tpf1(t), where f1(t) ∈ C([0, ∞[). Definition 2. A function f (t), t ≥ 0 is said to be in the space Cµn, µ ∈ IR, if f(n)∈ Cµ.
Definition 3. The Riemann-Liouville fractional integral operator of order α ≥ 0, for a function f ∈ Cµ, (µ ≥ −1) is defined as
Jαf (t) = Γ(α)1 Rt 0(t − τ )α−1f (τ )dτ ; α > 0, t > 0, J0f (t) = f (t), (6) where Γ(α) :=R∞ 0 e −uuα−1du.
For the convenience of establishing the results, we give the semigroup property: JαJβf (t) = Jα+βf (t), α ≥ 0, β ≥ 0, (7) which implies the commutative property
JαJβf (t) = JβJαf (t). (8) More details, one can consult [13].
3. Main Results
Theorem 3.1. Let p, q be two positive integrable functions on [0, ∞[ and let f and g be two differentiable functions on [0, ∞[. If f0, g0 ∈ L∞([0, ∞[), then for all t > 0, α > 0, we have:
J αq(t)Jαpf g(t) + Jαp(t)Jαqf g(t) − Jαqf (t)Jαpg(t) − Jαpf (t)Jαqg(t) ≤ ||f0||∞||g0||∞ h Jαq(t)Jαt2p(t) + Jαp(t)Jαt2q(t) − 2(Jαtq(t))(Jαtp(t))i. (9)
Proof. Let f and g be two functions satisfying the conditions of Theorem 3.1 and let p, q be two positive integrable functions on [0, ∞[.
Define
H(τ, ρ) := (f (τ ) − f (ρ))(g(τ ) − g(ρ)); τ, ρ ∈ (0, t), t > 0. (10) Multiplying (10) by (t−τ )Γ(α)α−1p(τ ); τ ∈ (0, t) and and integrating the resulting iden-tity with respect to τ from 0 to t, we obtain
1 Γ(α) Z t 0 (t − τ )α−1p(τ )H(τ, ρ)dτ = Jαpf g(t) − f (ρ)Jαpg(t) − g(ρ)Jαpf (t) + f (ρ)g(ρ)Jαp(t). (11)
Multiplying (11) by (t−ρ)Γ(α)α−1q(ρ); ρ ∈ (0, t) and integrating the resulting identity with respect to ρ over (0, t), we can write
1 Γ2(α) Z t 0 Z t 0 (t − τ )α−1(t − ρ)α−1p(τ )q(ρ)H(τ, ρ)dτ dρ = Jαq(t)Jαpf g(t) − Jαqf (t)Jαpg(t) − Jαpf (t)Jαqg(t) + Jαp(t)Jαqf g(t). (12)
On the other hand, we know that H(τ, ρ) = Z ρ τ Z ρ τ f0(y)g0(z)dydz, (13)
From the hypothesis f0, g0 ∈ L∞([0, ∞[), it follows |H(τ, ρ)| ≤ Z ρ τ f0(y)dy Z ρ τ g0(z)dz ≤ ||f 0|| ∞||g0||∞(τ − ρ)2. (14) Hence, 1 Γ2(α) Z t 0 Z t 0 (t − τ )α−1(t − ρ)α−1p(τ )q(ρ)|H(τ, ρ)|dτ dρ ≤ ||f0||∞||g0||∞ Γ2(α) Rt 0 Rt 0(t − τ )α−1(t − ρ)α−1(τ2− 2τ ρ + ρ2)p(τ )q(ρ)dτ dρ. (15)
Thus, we obtain the following inequality 1 Γ2(α) Z t 0 Z t 0 (t − τ )α−1(t − ρ)α−1p(τ )q(ρ)|H(τ, ρ)|dτ dρ ≤ ||f0||∞||g0||∞ h Jαq(t)Jαt2p(t) − 2(Jαtp(t))(Jαtq(t)) + Jαp(t)Jαt2q(t)i. (16)
By the relations (12), (16) and using the properties of the modulus, we get the desired inequality (9).
Remark 3.2. Applying Theorem 3.1 for α = 1, p(x) = q(x); x ∈ [0, ∞[, we obtain the inequality (2) on [0, t].
The second result is the following theorem.
Theorem 3.3. Let p, q be two positive integrable functions on [0, ∞[ and let f, g be two differentiable functions on [0, ∞[. If f0, g0 ∈ L∞([0, ∞[), then for all t > 0, α > 0, β > 0, we have J αp(t)Jβqf g(t) + Jβq(t)Jαpf g(t) − Jαpf (t)Jβqg(t) − Jβqf (t)Jαpg(t) ≤ ||f0||∞||g0||∞ h Jαp(t)Jβt2q(t) − 2(Jαtp(t))(Jβtq(t)) + Jβq(t)Jαt2p(t)i. (17)
Proof. The relation (11) implies that 1 Γ(α)Γ(β) Z t 0 Z t 0 (t − τ )α−1(t − ρ)β−1p(τ )q(ρ)H(τ, ρ)dτ dρ = Jβq(t)Jαpf g(t) + Jαp(t)Jβqf g(t) − Jβqf (t)Jαpg(t) − Jαpf (t)Jβqg(t). (18)
From the relation (13), it follows 1 Γ(α) Z t 0 (t − τ )α−1p(τ )|H(τ, ρ)|dτ ≤ ||f0||∞||g0||∞ h Jαt2p(t) − 2ρJαtp(t) + ρ2Jαp(t)i. (19)
The inequality (19) implies that 1 Γ(α)Γ(β) Z t 0 Z t 0 (t − τ )α−1(t − ρ)β−1p(τ )q(ρ)|H(τ, ρ)|dτ dρ ≤ ||f0|| ∞||g0||∞ h Jβq(t)Jαt2p(t) − 2(Jαtp(t))(Jβtq(t)) + Jαp(t)Jβt2q(t)i. (20)
Using (18), (20) and the properties of modulus, we finally obtain (17). Remark 3.4 (i) Applying Theorem 3.3 for α = β we obtain Theorem 3.1.
(ii) Applying Theorem 3.3 for α = β = 1, p(x) = q(x), x ∈ [0, ∞[, we obtain the inequality (2) on [0, t].
Theorem 3.5.Let p, q be two positive integrable functions on [0, ∞[ and let f, g be two integrable functions on [0, ∞[ satisfying the condition (4) on [0, ∞[. Then for all t > 0, α > 0, we have: J αq(t)Jαpf g(t) + Jαp(t)Jαqf g(t) − Jαqf (t)Jαpg(t) − Jαqf (t)Jαpg(t) ≤ MhJαp(t)Jαqg2(t) + Jαq(t)Jαpg2(t) − 2Jαpg(t)Jαqg(t)i. (21)
Proof. Let f and g be two functions satisfying the condition (4) on [0, ∞[. Then for every τ, ρ ∈ [0, t]; t > 0, we have
|f (τ ) − f (ρ)| ≤ M |g(τ ) − g(ρ)|. (22) This implies that
H(τ, ρ) ≤ M g(τ ) − g(ρ)2, τ, ρ ∈ [0, t]. (23) Hence, it follows that
1 Γ(α) Z t 0 (t − τ )α−1p(τ )|H(τ, ρ)|dτ ≤ MJαpg2(t) − 2g(ρ)Jαpg(t) + g2(ρ)Jαp(t). (24) Consequently, 1 Γ2(α) Z t 0 Z t 0 (t − τ )α−1(t − ρ)α−1p(τ )q(ρ)|H(τ, ρ)|dτ dρ ≤ MhJαq(t)Jαpg2(t) − 2Jαqg(t)Jαpg(t) + Jαp(t)Jαqg2(t)i. (25)
Using (12) and (25), we finally get the desired fractional inequality (21).
Remark 3.6 Applying Theorem 3.5 for α = 1, p(x) = q(x); x ∈ [0, ∞[, we obtain the inequality (5) on [0, t].
Another estimation depending on two fractional parameters is the following the-orem:
Theorem 3.7. Let f and g be two integrable functions on [0, ∞[ satisfying the con-dition (4) on [0, ∞[ and let p, q be two positive integrable functions on [0, ∞[. Then the inequality J αp(t)Jβqf g(t) + Jβq(t)Jαpf g(t) − Jαpf (t)Jβqg(t) − Jβqf (t)Jαpg(t) ≤ MhJβq(t)Jαpg2(t) + Jαp(t)Jβqg2(t) − 2Jαpg(t)Jβqg(t)i. (26)
is valid for all all t > 0, α > 0, β > 0. Proof. Using the relation (24), we obtain
1 Γ(α)Γ(β) Z t 0 Z t 0 (t − τ )α−1(t − ρ)β−1p(τ )q(ρ)|H(τ, ρ)|dτ dρ ≤ M Γ(β) Rt 0 (t − ρ)β−1q(ρ)hJαpg2(t) − 2g(ρ)Jαpg(t) + g2(ρ)Jαp(t)idρ. (27) Consequently, 1 Γ(α)Γ(β) Z t 0 Z t 0 (t − τ )α−1(t − ρ)β−1p(τ )q(ρ)|H(τ, ρ)|dτ dρ ≤ MhJβq(t)Jαpg2(t) − 2Jβqg(t)Jαpg(t) + Jβqg2(t)Jαp(t)i. (28)
Theorem 3.7 is thus proved.
Remark 3.8. (i) Applying Theorem 3.7 for α = β, we obtain Theorem 3.5. (ii) Applying Theorem 3.7 for α = β = 1, p(x) = q(x); x ∈ [0, ∞[, we obtain the inequality (5) on [0, t].
Acknowledges: This paper is supported by l’ ANDRU, Agence Nationale pour le Developpement de la Recherche Universitaire and UMAB University of Mosta-ganem, ( PNR Project 2011-2013).”
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Zoubir DAHMANI,
Laboratory LMPA, Faculty SESNV, UMAB, University of Mostaganem Mostaganem, Algeria
